Period mapping

A mapping which assigns to a point $ s $ of the base $ S $ of a family $ \{ X _{s} \} $ of algebraic varieties over the field $ \mathbf C $ of complex numbers the cohomology spaces $ H ^{*} (X _{s} ) $ of the fibre over this point, provided with a Hodge structure. The Hodge structure thus obtained is considered as a point in the moduli variety of Hodge structures of a given type.

The study of period mappings dates back to the studies of N.H. Abel and C.G.J. Jacobi on integrals of algebraic functions (see Abelian differential). However, until recently, the only period mappings that have been studied were those which correspond to families of curves.

Let $ \{ X _{s} \} $ be the family of fibres $ X _{s} = f ^{ {\ } -1} (s) $ of a smooth projective morphism $ f: \ X \rightarrow S $, where $ S $ is a smooth variety. The cohomology spaces $ H ^{*} (X _{s} ,\ \mathbf Z ) = V _{ {\mathbf Z}} $ are then provided with a pure polarized Hodge structure, which is defined by a homomorphism of real algebraic groups (cf. Algebraic group) $ h: \ \mathbf C ^{*} \rightarrow G _{ {\mathbf R}} $, where $ \mathbf C ^{*} $ is the multiplicative group of the field of complex numbers, considered as a real algebraic group, while $$ G = \{ {g \in \mathop{\rm GL}\nolimits (V)} : {\psi (gx,\ gv) = \lambda (g) \psi (x,\ y)} \} $$ is the algebraic group of linear transformations of a space $ V $ that multiply a non-singular (symmetric or skew-symmetric) bilinear form $ \psi $ by a scalar factor; the automorphism $ \mathop{\rm Ad}\nolimits \ h(i) $ of $ G _{ {\mathbf R}} $ is thus a Cartan involution and $ h( \mathbf R ^{*} ) $ lies in the centre of $ G _{ {\mathbf R}} $. The set $ X _{G} $ of homomorphisms $ h: \ \mathbf C ^{*} \rightarrow G _{ {\mathbf R}} $ which possess the above properties is naturally provided with the $ G _{ {\mathbf R}} $- invariant structure of a homogeneous Kähler manifold and is called a Griffiths variety, while the quotient variety $ M _{G} = X _{G} /G _{ {\mathbf Z}} $ is the moduli space of the Hodge structures. The homomorphism $ h $ defines the Hodge decomposition $$ \mathfrak G _{ {\mathbf C}} = \oplus \mathfrak G ^{p,-p} $$ of the Lie algebra $ \mathfrak G $ of the group $ G $, where $ \mathfrak G ^{p,-p} $ is the subspace in $ \mathfrak G _{ {\mathbf C}} $ on which $ \mathop{\rm Ad}\nolimits \ h(z) $ operates by multiplication by $ \overline{z} {} ^{p} z ^{-p} $. The assignment $ h \rightarrow P(h) $, where $ P(h) $ is the parabolic subgroup in $ G _{ {\mathbf C}} $ with Lie algebra $ \oplus _{ {p} \geq 0} \mathfrak G ^{p,-p} $, defines an open dense imbedding of the variety $ X _{G} $ into the compact $ G _{ {\mathbf C}} $- homogeneous flag manifold $ X _{G} $. In the tangent space $$ \mathfrak G _{ {\mathbf G}} / \oplus _ {p \geq 0} \mathfrak G ^{p,-p} $$ to $ X _{G} $ at the point $ h $, the horizontal subspace $$ \oplus _ {p \geq -1} \mathfrak G ^{p,-p} / \oplus _ {p \geq 0} \mathfrak G ^{p,-p} $$ is distinguished. A holomorphic mapping into $ X _{G} $ or $ M _{G} $ is said to be horizontal if the image of its tangential mapping lies in a horizontal subbundle.

It has been established that the period mapping $ \Phi : \ S \rightarrow M _{G} $ is horizontal (see , [3]). The singularities of period mappings are described by the Schmid nilpotent orbit theorem, which, when $ S = \overline{S} \setminus \{ 0 \} $ is a curve with a deleted point, asserts that if $ z $ is the local coordinate on $ S $, $ z(0) = 0 $, then when $ z \rightarrow 0 $, $ \Phi (z) $ is asymptotically close to $$ \mathop{\rm exp}\nolimits \left ( \frac{ \mathop{\rm log}\nolimits \ z}{2 \pi i} N \ \right ) a, $$ where $ a \in X _{G} $ and $ N \in \mathfrak G _{ {\mathbf Q}} $ is a nilpotent element (see [4]). The image of the monodromy group $$ \Phi _{*} ( \pi _{1} (S,\ s)) \subset G _{ {\mathbf Z}} $$ is semi-simple in every rational representation of the group $ G $, while transference of $ T $ around a divisor with normal intersections $ \overline{S} \setminus S $ in a smooth compactification $ \overline{S} $ of the variety $ S $ generates quasi-unipotent elements $ \Phi _{*} (T) \in G _{ {\mathbf Z}} $( i.e. elements which take roots of unity as eigen values). The importance of the monodromy group is underlined by the rigidity theorem (see , [2], [4]): If there are two families of algebraic varieties over $ S $, then the relevant period mappings $ \Phi _{1} $ and $ \Phi _{2} $ from $ S $ into $ M _{G} $ coincide if and only if $ \Phi _{1} (s _{0} ) = \Phi _{2} (s _{0} ) $ at a certain point $ s _{0} \in S $, and if the homomorphisms $ \Phi _{i\star} : \ \pi _{1} (S,\ s _{0} ) \rightarrow G _{ {\mathbf Z}} $, $ i = 1,\ 2 $, coincide.

Complete results on the structure of the kernel and the image of a period mapping generally relate to the cases of curves and $ K3 $- surfaces (cf. $ K3 $- surface). If $ \{ X _{s} \} $ is a family of varieties of the type indicated and $ \Phi (s) = \Phi (s ^ \prime ) $, then $ X _{s} \widetilde \rightarrow X _{ {s} ^ \prime } $( Torelli's theorem), while for $ K3 $- surfaces the maximum possible image of the period mapping coincides with $ M _{G} $( see [7]). In the case of curves, the image of the period mapping has been described partially (Schottky–Yung relations, see [6], [8]). The Griffiths conjecture states that a moduli variety permits a partial analytic compactification, i.e. an open imbedding in an analytic space $ \overline{M} _{G} $ such that the period mapping $ S \rightarrow M _{G} $ can be continued to a holomorphic mapping $ \overline{S} \supset S $ for every smooth compactification $ \overline{S} \supset S $. Such a compactification is known (1983) only for the case where $ X _{G} $ is a symmetric domain [9].

References

[1a]	P.A. Griffiths, "Periods of integrals on algebraic manifolds I" Amer. J. of Math. , 90 (1968) pp. 568–625 MR0229641 Zbl 0169.52303
[1b]	P.A. Griffiths, "Periods of integrals on algebraic manifolds II" Amer. J. of Math. , 90 (1968) pp. 805–865 MR0233825 Zbl 0183.25501
[1c]	P.A. Griffiths, "Periods of integrals on algebraic manifolds III" Publ. Math. IHES , 38 (1970) pp. 125–180 MR0282990 Zbl 0212.53503
[2]	P.A. Griffiths, "A transcendental method in algebraic geometry" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 113–119 MR0414551 Zbl 0227.14008
[3]	P. Deligne, "Travaux de Griffiths" , Sem. Bourbaki Exp. 376 , Lect. notes in math. , 180 , Springer (1971) pp. 213–235 Zbl 0208.48601
[4]	W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" Invent. Math. , 22 (1973) pp. 211–319 MR0382272 Zbl 0278.14003
[5]	E.H. Cattani, A.G. Kaplan, "Existence of period mappings for Hodge structures of weight two" Duke Math. J. , 4 : 1 (1977) pp. 1–43
[6]	B.A. Dubrovin, "Theta-functions and non-linear equations" Russian Math. Surveys , 36 : 2 (1981) pp. 11–92 Uspekhi Mat. Nauk , 36 : 2 (1981) pp. 11–80 MR616797 Zbl 0549.58038 Zbl 0478.58038
[7]	V.A. Kulikov, Uspekhi Mat. Nauk , 32 : 4 (1977) pp. 257–258
[8]	D. Mumford, Matematika , 17 : 4 (1973) pp. 34–43 MR0360082 MR0339966
[9]	W. Baily, A. Borel, "Compactification of arithmetic quotients of bounded symmetric domains" Ann. of Math. , 84 (1966) pp. 442–528 MR0216035 Zbl 0154.08602

Comments

In the one-variable case, W. Schmid also proved a very precise description of the asymptotics of the period mapping, the $ \mathop{\rm SL}\nolimits _{2} $- orbit theorem. This was generalized to the several-variables case by E. Cattani and A. Kaplan [a1]. The period mapping and period domain also have been considered in the case of singularities [a1], [a2]. See also Variation of Hodge structure.

References

[a1]	E. Cattani, A. Kaplan, "Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure" Invent. Math. , 67 (1982) pp. 101–115 MR0664326 Zbl 0516.14005
[a2]	E. Looijenga, "A period mapping for certain semi-universal deformations" Compos. Math. , 30 (1975) pp. 299–316 MR0376666 Zbl 0312.14006
[a3]	A.N. Varchenko, "Mapping of periods and intersection form" Funct. Anal. Appl. , 16 : 2 (1982) pp. 83–93 Funkts. Anal. Prilozhen. , 16 : 2 (1982) pp. 7–20 Zbl 0513.32018
[a4]	C. Peters, J.H.M. Steenbrink, "Infinitesimal variations of Hodge structures" K. Ueno (ed.) , Classical Algebraic and Analytic Manifolds , Birkhäuser (1984) pp. 399–463 MR0728615
[a5]	P. Griffiths, "Variation of Hodge structures" P. Griffiths (ed.) , Topics in Transcendental Algebraic geometry , Princeton Univ. Press (1984) pp. 3–28 MR0756854 MR0756846 MR0756845 MR0756843
[a6]	P. Griffiths, W. Schmid, "Recent developments in Hodge theory" , Discrete Subgroups of Lie Groups and Applications to Moduli , Oxford Univ. Press (1975) pp. 31–128 MR0419850 Zbl 0355.14003
[a7]	P. Griffiths, "Some transcendental aspects of algebraic geometry" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 3–110 MR0419438 Zbl 0309.14007
[a8]	J. Carlson, N.Green, P. Griffiths, J. Harris, "Infinitesimal variations of Hodge structure I" Compos. Math. , 50 (1983) pp. 109–205 MR0720288 MR0720289 MR0720290 Zbl 0531.14006
[a9]	R. Donagi, "Generic Torelli for hypersurfaces" Compos. Math. , 50 (1983) pp. 325–353 MR720291 Zbl 0598.14007
[a10]	M. Green, "The period maps for hypersurface sections of high degree of an algebraic variety" Compos. Math. , 55 (1984) pp. 135–156
[a11]	D.A. Cox, "Generic Torelli and infinitesimal variation of Hodge structure" S.J. Bloch (ed.) , Algebraic geometry (Bowdoin, 1985) , Proc. Symp. Pure Math. , 46 , Amer. Math. Soc. (1987) pp. 235–246 MR0927983 Zbl 0637.14008